Let $X$, $Y$, and $Z$ be points on a circle. Let $\overline{XY}$ and the tangent to the circle at $Z$ intersect at $W$. If $WX = 4$, $WZ = 8$, and $\overline{WY} \perp \overline{WZ}$, then find $YZ$.
[asy] unitsize(2 cm); pair A, B, C, D; D = (0,1); B = dir(150); C = dir(210); A = (-sqrt(3)/2,1); draw(Circle((0,0),1)); draw(D--A--C); label("$W$", A, NW); label("$X$", B, SE); label("$Y$", C, SW); label("$Z$", D, N); label("$8$", (A + D)/2, N); label("$4$", (A + B)/2, W); [/asy]
